If there's one game out there to really recommend the PlayStation Vita, it's probably Gravity Rush. Not a sequel, not a tie-in or a remake, it's barely even from a pre-existing genre. It's the sort of meaty, heavyweight game that you won't find on your iPhone or Android device.
The game stars a young amnesiac woman who we soon discover is named Kat. Kat awakes in a strange city in which things are getting decidedly weird. An infestation of aliens called the Nevi has broken out, but to make matters far worse gravity has gone to pot. Fortunately, thanks to a cat she discovers (an actual, four-legged cat with a 'c' this time), Kat can manipulate gravity, making her able to get to places and do things others can't.
It's up to the player to use these gravity bending powers to deal with the Nevi, rescue townfolk in danger of floating into oblivion and deal with a local crimelord.
Not only are the gameplay mechanics refreshingly outlandish the visuals are, too. The game's creator is Keiichiro Toyama, one of the men behind the original Silent Hill. While you'd be pushed to say the two games are overly similar, they do both veer into very strange territory. It's very clear in the anime-infused art style, which also riffs on the style of seminal French comics artist Moebius.
If you're looking for an experience you won't find anywhere but on your PSV, Gravity Rush is a great place to start.